Simplify the following expression and state the condition under which the simplification is valid. You can assume that $p \neq 0$. $x = \dfrac{p}{4(4p - 5)} \div \dfrac{5p}{5(4p - 5)} $
Answer: Dividing by an expression is the same as multiplying by its inverse. $x = \dfrac{p}{4(4p - 5)} \times \dfrac{5(4p - 5)}{5p} $ When multiplying fractions, we multiply the numerators and the denominators. $x = \dfrac{ p \times 5(4p - 5) } { 4(4p - 5) \times 5p } $ $ x = \dfrac{5p(4p - 5)}{20p(4p - 5)} $ We can cancel the $4p - 5$ so long as $4p - 5 \neq 0$ Therefore $p \neq \dfrac{5}{4}$ $x = \dfrac{5p \cancel{(4p - 5})}{20p \cancel{(4p - 5)}} = \dfrac{5p}{20p} = \dfrac{1}{4} $